On the chromatic number, colorings, and codes of the ... · since it is a distance-regular graph and because of the codes which it produces. With respect to J(n, w) we will consider

DISCRETE APPLlED

EISEVIER Discrete Applied Mathematics 70 (1996) 163-175 MATHEMATICS

On the chromatic number, colorings, and codes of the Johnson graph

Tuvi Etzion a,*, Sara Bitan b

aDepartment of Computer Science, Royal Holloway, University of London, Surrey TW20 OEX, United Kingdom

b Computer Science Department, Technion, Israel Institute of Technology, Haifa 32000. Israel

Received 14 October 1994; accepted 25 September 1995

Abstract

We consider the Johnson graph J(~,w), 0 < w < n. The graph has (E) vertices representing

the (z) w-subsets of an n-set. Two vertices are connected by an edge if the intersection between their w-subsets is a (w - I)-set. Let O(n,w) be the chromatic number of this graph. It is well known that O(n, w) f n. We give some constructions which yield Q(n, w) < n for some cases of n and w. The colorings associated with the chromatic number and other colorings of the graph lead to improvements on the lower bounds on the sizes of some constant weight codes.

1. Introduction

The Johnson graph J(n,w) is defined as follows. The vertex set, I’,$ consists of all

w-subsets of a fixed n-set (or all binary n-tuples with constant weight w). Two such

w-subsets (n-tuples) are adjacent if and only if their intersection has size w - 1 (there

are w- 1 coordinates in which both n-tuples have ONES). This graph is very interesting

since it is a distance-regular graph and because of the codes which it produces. With

respect to J(n, w) we will consider two fundamental questions in graph theory, the

chromatic number of the graph and its largest independent set. These problems can

be translated into coding theory and block design. A maximum independent set is the

largest code of length n, constant weight w, and minimum Hamming distance (distance

in short) 4 [4, 151. It is also the largest packing of (w - 1)-subsets by w-subsets. The

chromatic number of the graph is the minimum number of disjoint constant weight

codes of length n, weight w, and distance 4, for which the union is the set of all

n-tuples with weight w [4, 151. It is also the minimum number of disjoint packings of

(w- 1)-subsets by w-subsets, for which the union is the set of all w-subsets of the n-set.

* Correspondence address: Computer Science Department, Technion, Israel Institute of Technology, Haifa 3200, Israel. E-mail: [email protected].

This research was supported in part by the SERC of United Kingdom under grant no. GRK01605. The

author is on leave of absence from the Computer Science Dept., Technion.

0166-218x/96/$15.00 0 1996 Elsevier Science B.V. All rights reserved SSDZ 0166-218X(96)00104-2

164 i? E&ion, S. BitanIDiscrete Applied Mathematics 70 (1996) 163-175

These partitions of packings are used to produce other independent sets in a method

called the partitioning construction [4, 151. All these questions are of considerable

interest and we will try to give some answers now.

Let (n, d, w) denote a code of length n, constant weight w, and distance 4, and let

A(n,d, w) denote the maximum size of an (n,d, w) code. For bounds on A(n,d, w)

the reader is referred to [4, lo]. Let 0(n, w) denote the chromatic number of J(n, w). Graham and Sloane [lo] proved that 0(n, w) < n for all 0 < w < n. Trivially we

have B(n,w) = 0(n,n - w), e(n,O) = 1, and e(n, 1) = n [4, 151 . Also, it is easily

observed that for even n, 8(n,2) = n - 1 and for odd n, B(n,2) = n. For w = 3,

it is known [ll, 12, 141, that for n > 7, n E 1 or 3 (mod 6) B(n,3) = n - 2, and

for n > 7, n E 0 or 2 (mod 6), B(n,3) = II - 1, and 8(7,3) = 6, B(n,3) = n

for n < 6. For n E 4 (mod 6) we have 8(n,3) = n, and it is conjectured that for

n > 5, n 3 5 (mod 6), 0(n,3) = n - 1, but this is proved [5, 61 only for some

infinite cases. For w > 3 the evaluation of e(ti, w) becomes more difficult. van Pul

and Etzion [15] proved that if e(n, w) < n for all even w, then 8(2%, w) < 2’n for

all even w, and i 2 0. This result can be applied on n = 4,6, and 10 [ 15, 41. The

result proved in [15] is in fact that if for a given we, 8(n, w) < n for all even w,

2 < w < wo, then 8(2n,w) < 2n for all even w, 2 6 w d wo. For w = 4, clearly

8(2n, 4) < 2n implies 8(2%,4) < 2’n, i 2 1. This result can be applied on n =

2,3,5, and 7 [4]. Etzion [7] proved that if 0(2n,4) < 2n - 2, n E 2 or 4 (mod 6),

then 0(4n,4) < 4n - 2. This result was applied only to obtain 0(2’,4) < 2’ - 2, for

i 2 3.

Two types of constructions can be given for attaining new upper bounds on e(n,w),

direct constructions and recursive constructions. In Section 3, we will present a direct

approach, which unfortunately we could not apply without a computer search. We will

show cases for which e(n, w) < n - 1, when (w - 1 )w is relatively prime to n - 1. In

Section 4 we will give recursive constructions, a simple doubling construction and a

more complicated quintupling one which can be applied only on w = 4. In Section 5

we will discuss applications of the previous results. In Section 6 we will introduce a

specific interesting coloring of J( 11,4). In Section 7 we will improve the partitioning

construction for constructing independent sets in J(n, w) by using appropriate colorings.

But, we start in Section 2 with the definitions for the designs and methods used in this

paper.

2. The used designs and the partitioning construction

Since we use the partitioning construction to obtain new codes (larger independent

sets), and since we will improve this method we will first introduce the concept of

partitioning. The representation is taken from [4].

A partition ZI(n, w) = (XI , . . . ,X,) is a collection of disjoint sets or classes XI,. . . ,X,,

each of which is a code of length n, distance 4 and constant weight w, and whose

union contains all (z) vectors of weight w. The vector n(n, w) = (IX,/,. . . , IX,/) with

T E&on, S. Bitanl Discrete Applied Mathematics 70 (1996) 163-175 165

integer components is the index vector of the partitions Ii’(n,w), and

71(&W). n(n,w) = &q2

I=1

is its norm. We always assume IX11 > . . 2 IX, I. When there are several different

partitions available for a given n and w we often denote them by IIl(n, w), L’z(n, w), .

and their index vectors by z](n, w), z2(n, w), . . . .

The direct product L’(nl,wl)xLI(n2, ~2) of two partitions L’(n,,wl) = (Xl,. ,X,,),

II(n2, ~2) = ( YI, . . , Y,,,,) consists of the vectors

{(u,u): u E X,, v E Yj, 1 < i G m},

where m = min{ml, m2). This set (which is only part of the final code) clearly has

length nl + n2, distance 4, weight w1 + ~2, and contains

z(nl,wl). 4n2,w) = ~lKll&l i=l

words.

The construction: To obtain a code of length n, distance 4 and weight w by the

partitioning construction we write n = n1 + n2, choose E = 0 or 1, and take the union

of the direct products

Wnl,E) x U(n2,w - E),

fl(nl, E + 2) x IZ(n2, w - E - 2),

n(n,, E + 4) x U(n2,, w - E - 4),

It is apparent that this code does have the required properties, and contains

71(12,,&).71(n2,w-&)+7t(nl,&+2).7((n2,~-&-2)

+ n(nl, E + 4) .71(n2, w - e - 4) + f .

codewords. For examples of how the construction is applied to obtain specific codes

the reader is referred to [4, 151.

We next discuss the choice for a good partition. We say that one partition Ill(nl, WI )

dominates another II,(n,, WI ) if

~l(nl,wl)~ 4m,w2) 3 z~(~I,w). n(m,w2)

holds for all choices of n2, ~2, and all possible index vectors z(n2,wz). If a partition is

dominated it need never be used in the construction. As was proved by [4], nl(n,, WI) =

(al,... ,a,,) dominates 7c2(nl,wl) = (bl,... ,b,,) if and only if

i i cai > Cbi for all j = 1,. . . ,max{m1,m2}. i=l i=l

166 T. E&ion, S. Bitanl Discrete Applied Mathematics 70 (1996) 163-175

A partition n(n, w) is optimal if it dominates all other partitions n’(n, w) with the

same n and w.

We will also use in our paper some concepts of block design, and hence we give

some necessary definitions.

A Steiner system S(t, k,n) is a collection of k-subsets (called blocks) of an n-set

(whose elements are called points) such that each t-subset of the n-set is contained in

exactly one block. A packing quadruple system (PQ) of order n is a pair (Q, q), where

Q= {O,l,..., n - 1) is a set of points and q is a collection of 4-element subsets of Q

such that every 3-element subset of Q is a subset of at most one block of q. If every

3-element subset of Q is a subset of exactly one block of q the packing is a Steiner

quadruple system S(3,4,n). A packing triple system (PT) of order n is a pair (Q, q),

where Q= {O,l,..., n - I} is a set of points and q is a collection of 3-element subsets

of Q such that every 2-element subset of Q is a subset of at most one block of q. If

every 2-element subset of Q is a subset of exactly one block of q the packing is a

Steiner triple system S(2,3, n). A near-l-factorization of the complete graph K,,, n odd,

is a coloring of the graph edges with n colors. Each color consists of (n - 1)/2 edges

and one vertex is isolated. The set of edges of each color is called a near-l-factor. If

the vertex set of the graph is Z,,, then the near-l -factorization F = {Fo, FI, . . . , F,_ 1) is

a partition of pairs into n disjoint packings. It is well known that near-l-factorization

exists for every odd n.

A very interesting concept in block design is the large set. The large set is a partition

of the space into disjoint optimal designs. A near-l-factorization is a large set of near-

l-factors. For n f 1 or 3 (mod 6), n > 7, there exists a large set of Steiner triple

systems of order n, with n - 2 S(2,3,n) [I 1, 12, 141. For n 5 0 or 2 (mod 6), n > 7,

there exists a large set of PTs of order n, with n - 1 PTs which can be derived from

the large set of Steiner triple systems. Large sets of packing quadruple systems are not

known, except for trivial ones for n = 4 and n = 5, and the best results in this area

are given in [9].

3. A direct construction with computer search

In this section we will find partitions of all n-tuples with weight w into II - 1 codes

with minimum Hamming distance 4. This will be done by using computer search. To

make the search more effective, we must limit it, and this can be done by looking

for codes and partitions with a simple combinatorial and algebraic structure. We will

try to find partitions in which each code is obtained from the first code by a simple

permutation on the coordinates. To simplify the search even more, we will try that

the first code in the partition will have some nice automorphism, or “almost” a nice

automorphism.

T. Etzion. S. Bitanl Discrete Applied Mathematics 70 (1996) 163-175 167

Let C be an (n + 1, d,w) code. Each word c E C is represented by a w-tuple

c = (xi,xz, . . . ,xw), where Xi E {co} U 2,. For each binary word, c, of length n + 1

with constant weight w we define a shift operator Shi(c) by

sM(Xl,XZ ,...,&v)> = (~1 + i(mod n),. . .,x,,, + i(mod n)),

where 00 + i = co; for a code C, we define Shi(C) = {S/Q(C) : c E C}. The par-

tition will have the structure n(n + 1, w) = (&Cl,.. .,C,_i} where C, = Sh;(Co),

1 < i d n - 1. For n relatively prime to (w - 1 )w we define the layered graph Gnf ’ ( V, E),

with s layers, s = (“t’)/ n, where V = Us=, &, is the set of all w-subsets of {co} U Z,,

and E = {(u,u) 1 u,u E V, and Iu n UI = w - 1). All the layers Vi, 1 d i < s, are

disjoint, and are of equal size n. Each layer Vi consists of a single orbit of the oper-

ator Sk(*), on some w-subset of {oo} U Z,. Clearly an independent set in the graph

G”“( V, E) corresponds to an (n + 1,4, w) code. If, in addition to this, we find an

independent set C of size s in the graph that contains exactly one vertex from each

layer Vi, 1 < i < s, then the sets Ci = Sri(C), 0 d i < n, are also independent sets

in the graph, and {CO, Cl , . . . , &_I} is a partition of the vectors of length n + 1 and

weight w.

Let M(n) be the multiplicative group of the residues, between 1 and n - 1, modulo

n, which are relatively primes to n. For 0 E M(n), we define Z’g((xi, .,x,+,)) =

@XI,. . . , /?xw), where the multiplication is done modulo n, and Tg( vi) = { Z’p(u) : u E

Vi}. The graph Gi” = G”+l(Tp( V), E), where E is defined as before, is isomorphic

to G”+l( V,:E). Let Fb = { T,s, : i 2 0); Fp is a cyclic subgroup of the automorphism

group of the graph G”+‘( I’, E). Note that the order of 5b is equal to the order of p

in M(n). For each u E I’, let the orbit of u under Tg be [u]~ = {T(u) : T E Fp}.

We now turn to the description of our search program. The program receives three

parameters: an integer n, weight w, and /? E M(n). The program builds the graph

G”+l (I’, E), and using a backtracking algorithm it tries to build an independent set C

of size s, that contains exactly one vertex from each layer Vi. Let [v]; denote the

largest subset of [~]a for which, for each ui, uz E [VI!, we have &(u,) # uz for all

1 < k < n. In each step the program chooses a vertex u E V, where Vi is a layer that

still does not contain a vertex from the independent set, C, and checks if C U [u]; is

still an independent set. If it is, then [u]; is joined to C to form the new C. If for all

u E V; that we take, [u]g = [v]; then lJcEC B T (c) = C, i.e., T,s is an automorphism of

C. Clearly the complexity of the search decreases as the order of p in M(n) increases,

but p with too high order might not result in a code of size s. If you choose p = 1

then the search is the trivial search. The best choice (complexity wise) is to choose p

as a generator of M(n).

Using this search program we found several partitions. These partitions are listed in

the Appendix, together with /? and the order of p we used for the first code. Only for

n = 12 and w = 6, To is not an automorphism of C.

168 T. Etzion, S. Bitanl Discrete Applied Mathematics 70 (1996) 163-175

4. Recursive constructions

In this section we present two recursive constructions to obtain O(n, w) < n - 1. The

first one is a simple generalization of the result obtained in [ 151.

Theorem 1. Zf for a given wg, 6(n,w) < n for all 2 < w < WO, then 8(2n, w) < 2n

for all 2 < w < wa.

Proof. For even w the result was proven by [ 151 as mentioned in the Introduction. For

odd w note that if we take the union of ZI(n, 1) x n(n, w - 1) and n(n, w - 1) x n(n, 1)

as the first code, then by using all distinct direct products of the partitions to obtain

(2n, 4, w) codes we need no more than another 2n - 2 codes to cover all vectors of

length 2n and weight w. This is as in “Construction B” for combining partitions (see

[4, 151). 0

Let Ai, 0 < i < 4, denote a code of length 5 whose codewords have weight 4 in the

Lee metric, and uoaiu2asu4 is a codeword in Ai if and only if Cy=, jai E 4i (mod 5).

Let {PQo, . . . ,PQn-1) be a partition of quadruples on 2, U {a}. Let {PTo,. . . ,PT,_,}

be a partition of triples on 2, U {a}, and {Fo, . . ,F,_,} be a near-l-factorization of

K,, on Z,, such that in Fi, i E Z,, vertex i is isolated. From these sets we form the

following sets Sij, i E 25, j E Z,, on the points Z5 x Z, U {p}. For any given word

aoa1a2a3a4 E Ai we form in Sij one or two of the following block types:

(A) ((r,x),(s,y),(t,z),(q,x + Y +z +j)), ifa,=a,=a,=a,=l andallx,y,zEZ,.

(B) {G-,x), (i; .Y), (s, a), (s, w)>, if a, = a, = 2, and all {x,y} E F,,,, m E Z,, and {v,w} E Fm+j.

CC> {(i,x), (i, y), CO>, (6 w)), for any {x, y, z, w} E PQj.

CD) {(i,x), (i, y), C&z), PI,

for any {x, y, z, LX} E PQj.

(E) {G-,xh (r, u>, (r,z), 6, m +i>l, if a, = 3, a, = 1, and {x, y,z} E PT,,, for m E Z,,.

(F) I(r,x),(r,.v),(S,m +j),P], if a, = 3, a, = 1, and all {x, y, a} E PT, for m E Z,,.

(C) {G-,x), (r, u), (s, 0 (t, m + 2 + j)], ifa,=2,a,=a,=l,and(x,y)EF,,,form,IEZ,,.

(II) {(i,m),(s,O(t,m+ l+j),P], if ai = 2, a, = at = 1 and for all m, 1 E Z,,.

We leave the proof of the following theorem to the reader.

Theorem 2. {A’, : i E 5, j E Z,,} is a partition of all quadruples from a (5n + l)-set.

Corollary 1. Zf 0(n+ 1,4) <n for n E 1 or 5 (mod 6) 12 > 5, then B(5n+ 1,4) d 5n.

T. E&ion, S. Bitani Discrete Applied Mathematics 70 (1996) 163-l 75 169

To generalize this construction in order to obtain 8(kn + 1,4) d kn we need a set of

codes with weight 4 in the Lee metric, which satisfy certain properties. Unfortunately,

we are not optimistic about the existence of such codes, and hence we will not go into

details about this generalization.

5. Applications of the constructions

By using the partition of n(12,5), given in the Appendix and obtained via the

construction of Section 3, the partitions in [4] and the partitioning construction, four

new bounds on the A(n, 4, w) were obtained.

A(24,4,7) 2 15656.

A(24,4,9) > 59387.

A(24,4,11) 2 116937.

A(25,4,8) > 46832.

Another motivation to find partitions of quadruples with codes of the same size

is the suggested construction of Stanton [13] for producing Steiner quintuple system

S(4,5,2n + 1). For this construction a coloring of J(n + 1,4) with n colors, related to

n codes with the same size, is needed. Unfortunately, none of the partitions that we

have found was good enough to produce a new Steiner quintuple system.

Finally, we would like to mention that from the known results on 0(n,4) given in

the Introduction, from n( l&4) given in the Appendix, and from Corollary 1, we can

get many infinite sequences of values for n, for which 13(n,4) < n - 1. On the other

hand, Theorem 1 can be applied only on values of n where n E 0 (mod 6). The

reason for this is that for odd n, 8(n,2) = n, for IZ 3 4 (mod 6), B(n,3) = n, and for

IZ = 2 (mod 6), 2n E 4 (mod 6) and 8(2n,3) = n. By using 0(12,5) = 11, with its

coloring given in the Appendix, and the partitions for n = 12 given in [4] we obtain

that for 2 d w 6 10 we have 8(3 .2’, w) < 3 .2’ - 1 for i 3 2. Other results that can

be obtained are very specific, e.g., for 2 d w < 6 we have 0( 16, w) < 15.

6. An interesting partition of length 11 and weight 4

The following code is a constant weight code of length 11, weight 4, and minimum

Hamming distance 4,

100 100 100 01 001 110 000 01 100 001 010 10 100 010 110 00

010 010 010 01 001 000 110 01 010 011 000 10 010 001 011 00

001 001 001 01 000 100 010 11 010 000 110 10 001 100 101 00

100 011 000 01 000 010 001 11 001 101 000 10 010 110 100 00

100 000 011 01 000 001 130 11 001 000 011 10 001 011 010 00

010 101 000 01 010 100 001 10 100 110 000 10 100 101 001 00

010 000 101 01 001 010 100 10 100 000 101 10 000 101 110 00

170 T. E&ion, S. Bitanl Discrete Applied Mathematics 70 (1996) 163-175

000 110 011 00

000 011 101 00

011 010 001 00

101 001 100 00

110 100 010 00

110 000 000 11

By applying the eight permutations ( 1,2,3,6,4,5,&g, 7,10,11), ( 1,2,3,5,6,4,9.7,

8,10, ll), (9 7 8 1 2 3 4 5 6 lO,ll), (7,8,9,1,2,3,6,4,5,10,11), (8,9,7,1,2,3,5,6,4,10,11), 9 3 3 9 3 > 9 , 3 (4,5,6,8,9,7,1,2,3,10,1 l), (6,4,5 9 7 8 1 2 3 lO,ll), (5 6 4 7 8 9 1 2 3 lO,ll), on the first ,,>,,,, 3 9 , > 3 9 , 9 9 33 codewords we obtain 8 codes with 33 codewords for which we add the vectors

10100000011,01100000011,00011000011,00010100011,00001100011,00000011011,

00000010111, and 000000001111, respectively. Now, we have 9 disjoint codes of

length 11, weight 4, minimum distance 4, and size 34. The 24 missing vectors of

length 11 and weight 4 are the 24 vectors which contain the triples 11100000000,

00011100000, and 000000011100. These vectors can be easily partitioned to 8 disjoint

codes of size 3. This partition has norm 10476.

By exchanging some codewords between the codes we obtain a partition with in-

dex vector (34,34,34,34,34,34,34,34,34,12,3,3,3,3) and its norm is equal to 10584. Its

coloring is:

897CA325616445D136424556E5897839712214535664B67897289314978917823DBEC

A56314231297128945673978452316A235188642389715319A2123746452361231889

7641235A756127893431296931786459212437238A1528997578464562964A8575674

389641253123794531978268239756481127A3789563124A231853162943977868954

6485156437945A9678178894976556A744596286451897CABED312

where the words are ordered lexicographically and the list given is of their color’s

number [4]. This partition is of special interest for two reasons:

1. It leads to some new lower bounds on A(n, 4, w) as we will see in the next section.

2. Although A( 11,4,4) = 35 [l], usually the best lower bounds on A(n,4,4) for

n 3 5 (mod 6) are obtained by using the partitioning construction (other better bounds

are given in [2]). A code, with the same size of the one obtained by the partitioning

construction, is also derived by taking all codewords which start with 0 from a code

which attains A(n + 1,4,4) [3]. For iz = 11 and w = 4 the size of the derived code is

34. The maximum number of disjoint codes with this size is n - 2. So, in some sense

this partition is very close to a large set of quadruples. Unfortunately, the codes of

size 34 cannot be extended to codes which attain A( 12,4,4) = 51. Also, we could not

generalize this partition to obtain other partitions of length n = 5 (mod 6) with n - 2

codes like this.

As we will see in the next section, although the second partition dominates the first

one, the first one is more useful and it will be used in a modification of the partitioning

construction.

T. E&on, S. Bitanl Discrete Applied Mathematics 70 (1996) 163-175 171

7. Improving on the partitioning method

In this Section we will give some new bounds on A(n,4,w) which improve on the

ones that appeared in Brouwer et al. [4]. We use a method of improving on partitioning.

This method was applied on three parameters and we will present it with these specific

parameters.

Assume the partitioning method is applied on n = nl + n2. Assume further that n2

is odd and the direct products of II(nl,w) x Il(n2,O) and fl(q,w - 2) x Ii’(nz,2)

are taken. Usually, for IZ(nz,2) we use the optimal partition with n2 codes of size

(n2 - 1)/2. But without using the last coordinate, for the first n2 - 2 codes, we have

a partition with n2 - 2 codes of size (n2 - 1)/2 and n2 - 1 codes of size 1 with a

codeword which have a ONE in the last coordinate (usually we do not use these n2 - 1

codes in the offered method). Now, we can take the direct product of a code which

attains of length nl, weight w - 1, and minimum Hamming distance 4, by the word of

length n2, weight 1, with ONE in the last coordinate and join it to the code produced

by the partitioning construction. We must make sure that:

1. The union of the codes of weight w (taken for Z7(nl,w) x IZ(n,,O)) and w - 1 have

minimum Hamming distance 3.

2. The union of each of the codes with weight w - 2, if taken with direct products

with the codes of size 1, and the code of weight w - 1 have minimum Hamming

distance 3.

A similar idea can be applied if we use Il(nl, w - 3) x ZI(nz, 3) and we take Ii’(n2,3)

with codes for which some pair is not covered, and n2 - 2 codes of size 1 with

ONES in the uncovered coordinates (which again we usually do not use). A partition

like this can be obtained for n = 3k + 2, k 2 1, k odd. We were able to find a

partition of order 3k + 2 on the points Zsk+2 with 3k optimal packings, each one

does not cover the pair {3k, 3k + l}, and the remaining triples are {{i, k + i,2k + i} :

0 d i < k - l} U {{i,3k,3k + 1) : 0 < i f 3k - 1). The proof takes ideas given in

[5, 6, 81. {{i,k+ i,2k+ i} : 0 d i 6 k - 1) forms the (3k-t 1)th code. As an example,

for n = 11, the code

111000000 00 000110100 00 000101000 10 001001000 01

100000101 00 100001010 00 001000100 10 000000110 01

001100001 00 010100010 00 000000011 10 010000001 01

010001100 00 001010010 00 010010000 10 100010000 01

000011001 00

and the eight cyclic permutations on the first 9 coordinates result in 9 disjoint codes of

size 17. The remaining vectors are 10010010000, 01001001000, 00100100100, which

form the 10th code of the partition, and nine codes of size 1 with 2 ONES in the last

two coordinates.

For weight 4, we can use a similar idea if some triples are not covered. The partition

for n = 11, w = 4, and index vector (34,34,34,34,34,34,34,34,34,3,3,3,3,3,3,3,3) is good

172 T. Etzion, S. BitanIDiscrete Applied Mathematics 70 (1996) 163-175

for this purpose since the vectors 11100000000, 00011100000, 00000011100 are not

covered in the codes of size 34.

Now, we will present the six new bounds for n < 28 obtained by this method.

The partitions of pairs, triples, and quadruples discussed in this method will be called

special.

A(21,4,7) > 6156 with the following direct products. II(lO,O) x17(1 1,7), II(10,2) x

II(11,5), IZ(10,4) x II(11,3) (we take the special I7(11,3)), II(lO,6) x II(ll,l)

(we take IZ( 10,6) which attains 0(10,6) < 9), and the direct product of a code

which attains A( 10,4,5) by the vector of length 11 with 2 ONES in the last two

coordinates.

A(21,4,8) 2 10753 with the following direct products. II(lO,l)xI7( 11,7), IZ(10,3)x

17(11,5), L’(lO,5) x 27(11,3) (take the special ZI(11,3)), II(lO,7) x IZ(ll,l) (from

ZI(10,7) we use only 9 codes of size 13), and the direct product of a code which

attains A( 10,4,6) by the vector of length 11 with 2 ONES in the last two coordinates.

A(21,4,9) 2 16897 with the following direct products. I7( 10,2) x IZ( 11,7) (we take

the special 17(11,7)), IZ(10,4) x 17(11,5), L’(lO,6) x II(11,3) (we take the special

ZI( 11,3)), II( 10,8) x IZ( 1 1, 1 ), the direct product of a code which attains A( 10,4,7) by

the vector of length 11 with 2 ONES in the last two coordinates, the direct product of

the vector 0000000001 by the vectors 00011111111,11100011111,11111100011, and

the direct product of the vector 0000000000 by the vectors 0 110 1 1 1 1 1 1 1, 10 110 111111,

11011011111,11111101101,11111110110*

.4(22,4,7) 2 8252 with the following direct products. II( 11,l) x II( 11,6), ZI( 11,3) x

IZ(11,4), IZ(ll,5) x I7(11,2) (we take the special I7(11,2)). We take a code which

attains A( 12,4,7) = 80, delete its last coordinate to get a code A with weight 6, and a

code B with weight 7. Now we take the direct product of B by 00000000000, and the

direct product of A by the vector of length 11 with ONE in the last coordinate. Since

not all the vectors of weight 5 are covered by A we can use an appropriate permutation

such that we can add the codeword which is formed by a direct product of the last

and only unused codeword of the partition IT( 11,5) by the word 10000000001.

A(22,4,8) 2 16430 with the following direct products. II(11,7) x II(l1, 1) (for

II(11,7) we use only 9 codes of size 34 from the special Ii’(ll,7)), II(11,5) x

ZI( 11,3) (we take the special II(l1,3)), the direct product of a code which attains

A( 11,4,6) by the vector of length 11 with 2 ONES in the last two coordinates, and

from I7( 11,8) x L’( 11,O) take the direct product of the three uncovered vectors of

weight 8 in I7( 11,7) by the vector 00000000000. Similarly we take vectors from the

direct products I7(11,3) x IZ(11,5), I7(11,2) x I7(11,6), ZI(ll,l) x II(11,7), and

II(ll,O) x II(ll,8).

A(25,4,10) 2 140340 with the direct products as in Brouwer et al. [4], where in

IZ( 12,8) x I7( 13,2) we take the special II( 13,2), we add the direct product of a code

which attains A( 12,4,9) by the vector of length 13 with a ONE in the last coordinate,

and in II(12,lO) x II( 13,0) we take the only code of length 12, weight 10, and size

6, for which the union with the code which attains A( 12,4,9) is a code with minimum

distance 3.

T. E&ion, S. Bitanl Discrete Applied Mathematics 70 (1996) 163-175 173

Appendix

Let p E GF(p), p prime, and let r = o(j). For each partition we list a set of

words, {co,. . , c,_l } of weight w. II( p + 1, w) = {Co, Cl,. . . , C,_, } is obtained as

follows.

C = {PC, : 0 d i < S, and 0 < j < Y, flj’ci # Shk(/Y’ci), 0 Q jl < j2 < Y,

1 d k < P},

Ci = Shi(C) for 0 d i < p.

n(l2,4)

P = 4, o(P) = 5,

(co,O, 1,lO) (cx~,3,4,6) (~5,7,8)

(2,3,4,5) (4,5,6,g) (5,6,7,10)

(0, 13% 6) (L&3,9) (O,Z 3310)

n(l2,5)

P = 4, o(P) = 5,

(~0,1,2,3) (~3,4,5>7) (00,5,6,7,10)

(~0,1,6,7,8) (~0,1,7,10) (~1,2,4,5)

(0,1,8,9,10) (4,5,6,7,9) (0,1,2,5,10)

(3,7,8,9,10) (2,3,4,5,10) (3,5,6,7,8)

(0,2,3,9,10) (0,1,2,6,8)

(2,6,7,&10) (1,3,4,5,9)

P = 4, o(P) = 5,

(% 1,2,3,4,5) (00,1,7,8,9,10) (~0,1,2,5,10)

(m, 0,4,5,6,7) (00,0,5,g,%lO) (~3,5,6,7,8)

(00,5,6,7,9,10) (00,2,6,7,8,10) (~0,4,6,9,10)

(co, 1,2,3,7,10) (0,6,7,8,9,10) (0,1,2,4,9,10)

(1,5,6,7,8,9) (0,1,2,3,4,8) (0, 2,3,4,5,6)

(1,2,3,4,6,7) (3,4,5,6,8,10) (2,4,5,6,7,9)

(0, 1,2,4,5,7) (0,1,3,4,5,9)

174 T. E&ion, S. &tan/ Discrete Applied Mathematics 70 (1996) 163-I 7.5

W4,4)

B = 8, o(B) = 4,

(wO,l, 12) (m2,3,5)

(~7 0,6,7) (mO,2,11)

(8,9,10,12) (0,1,2,5)

(1,2,11,12) (3,4,6,8)

(4,7,8,10) (426,799)

(1,5,10,11) (3,6,7,10)

(1,3,10,12) (1,5,8,12)

W&4)

P = 8, o(P> = 8,

(cQ,O, 1316) (%2,3,5)

(%3> 427) (ml, 236)

(00,6,8,12) (5,6,7,8)

(0,7,8,9) (4,10,11,12)

(4,5,7,11) (0,6,10,11)

(4,5,8,9) (0,3,6,12)

(2,3,10,11) (2,6,7,11)

(w3,4,7) (00,4,15,16)

(m4,5,11) (m,O, 3714) (7,8,9,10) (4,5,6,8)

(10,11,12,15) (1,2,3,7) (9,10,11,16) (0,6,15,16)

(1,2,15,16) (O,ll, 12,14) (0,4,14,15) (4,5,7,14)

(1,2,4,12) (5,10,11,13) (3,4,6,16) (h&9,11)

(6,7,10,11) (0, 1,4,7) Cl,& 14915) (55% 9912)

(5,6,11,12) (2,7,13,14) (3,8,9,14) (2,8,9,15)

References

[l] M.R. Best, A(1 1,4,4) = 35 or some new optimal constant weight codes, Technical Report ZN 71/77,

Math. Centr. Amsterdam (1977).

[2] S. Bitan and T. Etzion, The last packing number of quadruples and cyclic SQS, Designs Codes

Cryptography 3 (1993) 283-313. [3] A.E. Brouwer, On the packing of quadruples without common triples, Ars Combin. 5 (1978) 3-6.

[4] A.E. Brouwer, J.B. Shearer, N.J.A. Sloane and W.D. Smith, A new table of constant weight codes,

IEEE Trans. Inform. Theory, IT-36 (1990) 13341380.

[5] T. Etzion, Optimal partition for triples, J. Combin. Theory Ser. A 59 (1992) 161-176.

[6] T. Etzion, Partitions of triples into optimal packings, J. Combin. Theory Ser. A 59 (1992) 269-284.

[7] T. Etzion, Partitions for quadruples, Am Combin. 36 (1993) 296-308.

[8] T. Etzion, Large sets of coverings, J. Combin. Designs, 2 (1994) 359-374.

[9] T. Etzion and A. Hartman, Towards a large set of Steiner quadruple systems, SIAM J. Discrete Math,

4 (1991) 182-195.

[lo] R. Graham and N. Sloane. Lower bounds for constant weight codes, IEEE Trans. Inform. Theory

IT-26 (1980) 3743.

[l I] J.X. Lu, On large sets of disjoint Steiner triple systems i-iii, J. Combin. Theory Ser. A 34 (1983)

140-146, 147-144, 156-183.

[12] J.X. Lu, On large sets of disjoint Steiner triple systems iv-vi, J. Combin. Theory Ser. A 37 (1984)

136163, 164188, 189-192.

T. Etzion, S. Bitanl Discrete Applied Mathematics 70 (1996) 163-I 75 175

[13] R.G. Stanton, A conjecture on quintuple systems, Ars Combin., 10 (1980) 187-192.

[14] L. Teirlinck, A completion of Lu’s determination of the spectrum for large sets of disjoint Steiner

systems, J. Combin. Theory Ser. A 57 (1991) 302-305.

[ 151 CL. van Pul and T. Etzion, New lower bounds for constant weight codes, IEEE Trans. Inform. Theory

IT-35 (1989) 132461329.